Basically, you have 3 options at the time of this writing:
WebSockets
WebSockets is a lightweight messaging protocol that utilizes TCP, rather than a Javascript implementation of TCP sockets, as you've noted. However, beyond the initial handshake, there are no HTTP headers being passed to and fro beyond that point. Once the connection is established, data passes freely, with minimal overhead.
Long-polling
Long-polling, in a nutshell, involves the client polling the server for new information periodically with HTTP requests. This is extremely expensive in terms of CPU and bandwidth, as you're sending a hefty new HTTP header each time. This is essentially your only option when it comes to older browsers, and libraries such as Socket.io use long-polling as a fallback in these cases.
WebRTC
In addition to what has been mentioned already, WebRTC allows for communication via UDP. UDP has long been used in multiplayer games in non web-based environments because of its low overhead (relative to TCP), low latency, and non-blocking nature.
TCP "guarantees" that each packet will arrive (save for catastrophic network failure), and that they will always arrive in the order that they were sent. This is great for critical information such as registering scores, hits, chat, and so on.
UDP, on the other hand, has no such guarantees. Packets can arrive in any order, or not at all. This is actually useful when it comes to less critical data that is sent at a high frequency, and needs to arrive as quickly as possible, such as player positions or inputs. The reason being that TCP streams are blocked if a single packet gets delayed during transport, resulting in large gaps in game state updates. With UDP, you can simply ignore packets that arrive late (or not at all), and carry on with the very next one you receive, creating a smoother experience for the player.
At the time of this writing, WebSockets are probably your best bet, though WebRTC adoption is expanding quickly, and may actually be preferable by the time you're done with your game, so that's something to consider.